    An important structural transformation that appears in many contexts, including homotopy associativity of algebraic structures [1]}, [2]}, pair of pants decomposition of Riemann surfaces [3]}, Feynman diagrams in physics (e.g., the four-point correlation in the \(s\)  and \(t\)  channels [4]}), Morse theory and singularity theory [5]}, [6]}, reconnection of vortices in quantum fluids [7]}, Lifshitz transitions of the Fermi volume [8]}, [9]}, [10]} and quantum transport of electrons in wire networks [11]}, [12]}, is given in figure REF .
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